Question

How many significant figures are there in (a) \(0.000054\) and (b) \(3.001 \times 10^{5}\) ?

Short answer

(a) 0.000054 has 2 significant figures. (b) 3.001 × 10^5 has 4 significant figures.

Step-by-step solution

Understand Significant Figures Definition

Significant figures are the digits in a number that contribute to its accuracy. They include all non-zero digits, any zeroes between significant digits, and any trailing zeroes in the decimal part.

Analyze the Number 0.000054

To find the significant figures in the number \(0.000054\), ignore the leading zeros. These zeros are only placeholders. The digits \(5\) and \(4\) are significant, so the number \(0.000054\) has 2 significant figures.

Analyze the Number 3.001 × 10^5

In scientific notation, only the digits before the multiplication sign \( \times \) and after the decimal point are considered for significant figures. The number \(3.001\) contains the digits \(3, 0, 0, 1\), all of which are considered significant. Therefore, \(3.001 \times 10^5\) has 4 significant figures.

Key concepts

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a concise form. It consists of two parts: a coefficient and an exponent. The coefficient is a number greater than or equal to 1 and less than 10, while the exponent shows the power of 10 by which the number is multiplied.
For instance, in the number \(3.001 \times 10^5\), \(3.001\) is the coefficient, and \(10^5\) indicates that the decimal point is moved five places to the right. This format is particularly useful in scientific calculations as it simplifies the process of working with extremities in numbers by focusing on the significant digits.
When using scientific notation, it’s key to identify significant figures only in the coefficient part. Zeros appearing in the exponent part do not affect significant figures. This simplification helps maintain accuracy without unnecessary complexity.

Accuracy in Measurements

Accuracy in measurements is about how close a measured value is to the true value or the standard. When reporting measurements, accuracy is reflected in the number of significant figures used. The more significant figures, the more precise a measurement tends to be.
It's important to understand that every measurement has an inherent degree of uncertainty, and significant figures help in expressing this uncertainty effectively. Measurements need to be reported in a way that communicates the confidence in the precision. For instance, a number like \(3.001\) with four significant figures signals more precision compared to a number like \(0.000054\) with only two significant figures.
In practice, when performing calculations, intermediate results should carry as many significant figures as possible to minimize rounding errors, and the final result should be rounded off to the appropriate number of significant figures as determined by the values involved.

Placeholder Zeros

Placeholder zeros are zeros that help position the decimal point in a number and are not considered significant. They serve to ensure that the decimal point appears in the correct position, acting essentially as place markers.
When evaluating significant figures, leading zeros, as seen in numbers like \(0.000054\), are not counted. These leading zeros are only there to show the number's location relative to the decimal and are not part of the measure of accuracy.
After a decimal point, trailing zeros can be significant if they occur at the end of a number, as they indicate precision. However, without a decimal point, these trailing zeros might not be significant. Understanding which zeros are just placeholders and which are part of the numeric precision is critical for correctly interpreting and communicating quantitative data.